Problem: Integrate. $\int\left(\dfrac2x-4e^x \right)dx=\,?$ Choose 1 answer: Choose 1 answer: (Choice A) A $2\ln|x|-4e^x+C$ (Choice B) B $2\ln|x|-e^{4x}+C$ (Choice C) C $2\ln(x)-e^{4x}+C$ (Choice D) D $2\ln(x)-4e^x+C$
Solution: We can integrate using the following formulas for the indefinite integrals of $e^x$ and $\dfrac1x$ : $\begin{aligned} &\int e^x\,dx=e^x+C \\\\ &\int \dfrac1x\,dx=\ln|x|+C \end{aligned}$ $\begin{aligned} &\phantom{=}\int\left(\dfrac2x-4e^x \right)dx \\\\ &=2\int \dfrac1x\,dx-4\int e^x \,dx \\\\ &=2\ln|x|-4e^x+C \end{aligned}$